Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 4 / d

Table of contents
- Solution d

Solution

For every stopping time

τ

, the martingale case of the optional sampling theorem for a supermartingale gives

E X_{τ} = X_{0} = 0

. Hence

E (Z_{τ} + X_{τ}) = E Z_{τ} .

Use the optimal stopping time from part c and the pathwise inequality

max_{0 \leq t \leq T} (Z_{t} + X_{t}) \geq Z_{τ^{*}} + X_{τ^{*}} .

Taking expectations gives

E max_{0 \leq t \leq T} (Z_{t} + X_{t}) \geq E Z_{τ^{*}} = U_{0} .

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